Probability

Board: CBSE Class: 12

Chapter Overview

Conditional probability P(A|B) = P(A n B)/P(B). Multiplication theorem: P(A n B) = P(A)P(B|A). Independent events: P(A n B) = P(A)P(B). Law of total probability and Bayes Theorem for reverse probability. Random variables assign numbers to outcomes. Probability distribution, mean (expected value), variance. Bernoulli trials and Binomial distribution.

Topics Covered

Conditional Probability
Multiplication Theorem
Independent Events
Bayes Theorem
Law of Total Probability
Random Variables
Probability Distribution
Mean and Variance
Bernoulli Trials
Binomial Distribution

Key Formulas

P(A|B) = P(A n B)/P(B)

P(A n B) = P(A)P(B) for independent

Bayes: P(A|B) = P(B|A)P(A)/P(B)

E(X) = sum xi pi

Var(X) = E(X²) - [E(X)]^2

P(X = k) = C(n,k) p^k q^n-k

Real-World Applications

Applications: Medical testing, spam filtering (Bayes), quality control, risk assessment, machine learning.

Study Tips

Tip: Define events carefully

Tip: Use probability trees for sequential events

Tip: Practice Bayes problems thoroughly